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The thread: What is conserved — page 33

Page 33 of 33, continuing through the 292 essays this motif runs through.

292 essays carry this thread — page 33 of 33.

The plateau everybody looks for is a summit, and a low one. −Dₗₗₗ/((4/5)εr) against separation in decaying turbulence at five Taylor-scale Reynolds numbers. None has a plateau at one. Each rises through the viscous range and turns over, peaking at 0.49, 0.63, 0.75, 0.85, 0.90 for Reλ = 50, 100, 200, 500, 1000. A measurement of ε that takes the largest value of this curve as four-fifths reads each of those shortfalls as a smaller dissipation.

The decay inside the four-fifths law

The four-fifths law gives the dissipation of turbulence from one measured moment with no constant in it, which makes it the obvious way to measure how the dissipation coefficient travels during a decay. But the law is exact only in a limit, and a decaying flow is not in it. The decay itself takes a quarter off the moment at the Reynolds numbers grids reach — and the bias moves as the flow decays, by a sixth, in the direction opposite to the effect being looked for.

$What every wing pays to bend its root less. Least induced drag against the root bending moment of the lift, both as fractions of the elliptic monoplane of the same span and lift, for the monoplane and for box wings with gaps of a tenth, a fifth and two-fifths of the span. Each curve is a parabola with its minimum at that wing's unconstrained optimum. The box wings' minima sit to the right of the monoplane's — they load their lift further out — and their parabolas are shallower: the gap-of-a-fifth box can bend its root 21 per cent less than the elliptic monoplane and still match its drag.$ Circulation and lift

A lighter spar turns a box wing into a biplane

Prandtl's best wing system has two-thirds of a monoplane's induced drag at a gap of a fifth of the span, and part of that saving is carried by circulation turning the corner into its fins. Ask the box to bend its root less and it pays about half what a monoplane pays — but a biplane with no fins pays nearly as little, and by the time the spar is a fifth lighter the fins carry almost nothing and the box has become the biplane it was built from.

At a fixed impulse every orbit is a curve of constant energy. Axial separation of two coaxial rings against the radius of the ring that started smaller, for six pairs with the same total impulse, all starting in one plane. Pairs starting nearer equal (inner curves) trace closed loops: the separation swings from one sign to the other as the rings take turns, and the radius swings with it. Pairs starting further apart run off to one side and never return. The pale curves are the separatrix, the energy of two equal free rings — it passes through the plane of the start at a ratio of 0.340 and its arms reach out to infinite separation.

Circulation and lift

Two rings leapfrog only if they start alike

Two coaxial smoke rings passing through each other in turn is the most famous thing vortices do, and it is not what two rings do in general. Set them off from one plane with different radii and below a definite ratio the smaller one draws ahead and never comes back. Which of the two happens is decided before either ring moves, by whether two separate rings could hold the pair's energy.

Three lobes, then a filament. An ellipse of aspect ratio 4 with a three-lobed bump of three thousandths, as contour dynamics carries it, drawn in the frame turning with the undisturbed ellipse at t = 0, 30 and 42. By t = 30 the bump has grown to a visible three-fold asymmetry — one end fattened, the other thinned — and by t = 42, about a turn and a tenth of the ellipse, the thinned end is being drawn out into a filament. The march is stopped there, while the area is still conserved to a few parts in a thousand; resolving the filament needs a contour that adds nodes, which this one does not.

Ideal flow

Past three, an ellipse is a shear layer

Kirchhoff's elliptical vortex turns for ever without changing shape, and Love showed in 1893 that it stops being stable at an aspect ratio of exactly three. Computed, that threshold turns out to be the first of a sequence — a new way of coming apart every one and a half aspect ratios — and the sequence ends somewhere recognisable. A long enough ellipse is a strip of vorticity, and it comes apart the way a shear layer does, at a rate Rayleigh found for the strip.

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